Optimal. Leaf size=77 \[ \frac {2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac {\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}-\frac {c^2 d^2}{2 e^3 (d+e x)^2} \]
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 43} \begin {gather*} \frac {2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac {\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}-\frac {c^2 d^2}{2 e^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^5} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^5}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^4}+\frac {c^2 d^2}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}+\frac {2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac {c^2 d^2}{2 e^3 (d+e x)^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.79 \begin {gather*} -\frac {3 a^2 e^4+2 a c d e^2 (d+4 e x)+c^2 d^2 \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 108, normalized size = 1.40 \begin {gather*} -\frac {6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 140, normalized size = 1.82 \begin {gather*} -\frac {{\left (6 \, c^{2} d^{2} x^{4} e^{4} + 16 \, c^{2} d^{3} x^{3} e^{3} + 15 \, c^{2} d^{4} x^{2} e^{2} + 6 \, c^{2} d^{5} x e + c^{2} d^{6} + 8 \, a c d x^{3} e^{5} + 18 \, a c d^{2} x^{2} e^{4} + 12 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + 3 \, a^{2} x^{2} e^{6} + 6 \, a^{2} d x e^{5} + 3 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{12 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 83, normalized size = 1.08 \begin {gather*} -\frac {c^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}-\frac {2 \left (a \,e^{2}-c \,d^{2}\right ) c d}{3 \left (e x +d \right )^{3} e^{3}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 108, normalized size = 1.40 \begin {gather*} -\frac {6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 85, normalized size = 1.10 \begin {gather*} -\frac {\frac {a^2\,e}{4}-d\,\left (\frac {c^2\,x^3}{3}-\frac {2\,a\,c\,x}{3}\right )-\frac {c^2\,e\,x^4}{12}+\frac {a\,c\,d^2}{6\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.97, size = 114, normalized size = 1.48 \begin {gather*} \frac {- 3 a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4} - 6 c^{2} d^{2} e^{2} x^{2} + x \left (- 8 a c d e^{3} - 4 c^{2} d^{3} e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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